Download Example 2 Second-Order Chemical reaction

MAT 3237 Section 2.8 B Handout
Classwork
1. Air Resistance
A differential equation describing the velocity v of a falling object subject to air
resistance proportional to the instantaneous velocity is
dv
m
 mg  kv
dt
where m is the mass of the object, and k  0 is a constant of proportionality. The
positive direction is downward.
(a) Solve the equation subject to the initial condition v(0)  v0 .
1
(b) Use the solution in part (a) to determine lim v(t ) , the limiting, or terminal,
t 
velocity of the object .
lim v(t ) 
t 
(c) If the distance s, measured from the point where the mass was released above
ground, is related to velocity v by
ds
v,
dt
find an explicit expression for s (t ) if s (0)  0 .
2
2. Drug Dissemination
A mathematical model for the rate at which a drug disseminates into the bloodstream
is given by
dx
 r  kx ,
dt
where r and k are positive constants. The function x(t ) describes the concentration of
the drug in the bloodstream at time t.
(a) Solve the DE subject to x(0)  0 .
3
(b) Find the limiting value of x(t ) as t   .
lim x(t ) 
t 
(c) At what time is the concentration one-half this limiting value?
Answers
1. (a) v(t ) 
mg 
mg   ktm
  v0 
e
k 
k 
(b)
kt
 
mg
mg
m
mg  
m
(c) s(t ) 
t   v0 
1

e


k
k
k
k 

4